1. No. Do not have to be from the same book/supplement. Everything in it must be from the same FACTION. Also note that it's still a CAD, just happens to be the one you chose as your primary and has your Warlord in it.
2. No. Do not have to be from the same book/supplement. Everything in it must be from the same FACTION.
3. No. Must be from a different FACTION than your primary.
Faction, not book, not codex, not supplement. Faction.
Your qualifications are kinda saying the same thing as mine, though I guess I forgot to add in that of course you can have stuff from Imperial Armour and other FW stuffs as well as the new tend to make online dataslates and WD articles and all that what is also the same Faction.
The FACTION normally = the Book / Supplement that you are taking from
The exception to this is of course the Space Marines book what says with the FAQ that the Chapters class as different factions.
Otherwise with the way you are saying it and the BRB says
"In the case of codex supplements, the Faction of all the units described in that publication is the same as the codex it is a supplement of."
This would mean I could as a Primary Detachment (the only difference between a CAD and Primary is the inclusion of the Warlord and the allied detachments the army can then take, that is why I made the difference between them) could have a Crimson Slaughter Chaos Lord with CS items, and the other HQ could be a CSM Chaos Lord with Axe of Blind Fury if you wanted it.